Book Summary

Chapter 1 is a historical overview of digital sound synthesis techniques--though far from complete, it highlights the link between abstract sound synthesis methods, based essentially on signal processing manipulations, and more modern physical modeling sound synthesis methods, as well as the connections among the various physical modeling methodologies. In Chapter 2, time series and difference operators are introduced, and some time is spent on the frequency domain interpretation of such operators, as well as on the development of certain manipulations which are of use in energy analysis of finite difference schemes. Special attention is paid to the correspondence between finite difference operations and simple digital filter designs. The simple harmonic oscillator is introduced in Chapter 3, and serves as a model for many of the systems which appear throughout the rest of the book. Various difference schemes are analyzed, especially with respect to numerical stability and accuracy, using both frequency domain and energetic principles; linear loss mechanisms are also introduced. Chapter 4 introduces various nonlinear excitation mechanisms in musical acoustics, many of which reduce to nonlinear generalizations of the harmonic oscillator, as well as associated finite difference schemes. Chapter 5 is designed as a reference chapter for the remainder of the book, with a complete introduction to the tools for the construction of finite difference schemes for partial differential equations in time and one or two spatial dimensions, including grid functions, difference operators, as well as a description of frequency domain techniques and inner product formulations, which are useful for nonlinear problems. As a test problem, the 1D wave equation and a variety of numerical methods are presented in Chapter 6. Various features of interest in musical simulations, including proper settings for boundary conditions, readout and interpolation, numerical dispersion and its perceptual significance, and numerical stability conditions are discussed. In addition, finite difference schemes are related to modal methods, digital waveguides and lumped networks, and relative strengths and weaknesses are evaluated. Chapter 7 deals with more musical extensions of the 1D wave equation and finite difference schemes to the case of vibrating bars of variable cross section, as well as stiff strings, and considerable time is spent on loss modelling as well as the coupling with hammer/mallet models and the special case of string preparation. The first serious foray into distributed nonlinear systems occurs in Chapter 8, with a discussion of nonlinear string vibration. Various models, of differing degrees of complexity are presented, and certain important perceptual effects of string nonlinearity, such as pitch glides and phantom partial generation are discussed. The construction of sound synthesis methods makes use of purely energetic techniques in this case. Chapter 9 picks up from the end of Chapter 6 to deal with linear wave propagation in acoustic tubes, which are the resonating elements in woodwind and brass instruments. Webster's equation, and finite difference methods are introduced, as are features of musical interest such as tome hole modelling, bell radiation, and coupling to reed-like excitation mechanisms, and the glottis in the case of vocal synthesis. Chapter 10 constitutes the first discussion of two-dimensional musical instruments, and in particular membranes and plates; it is directly analogous, in many respects, to Chapter 7. Special cases such as drum membranes, plate reverberation and piano soundboards are dealt with in detail. Mallet interaction, two-dimensional interpolation necessary for sound output, and direction-dependent numerical dispersion in finite difference schemes, as well as loss modeling are also discussed. Chapter 11 continues with the topic of two-dimensional vibration, in the nonlinear case, in order to deal with perceptually crucial effects such as crashes in instruments such as gongs and cymbals, and, as in Chapter 8, energy methods are developed. Finally, in Chapters and 13, some other numerical techniques with potential applications to musical sound synthesis, in particular finite element methods and spectral methods, are introduced. Concluding remarks appear in Chapter 14. Appendix A contains some simple Matlab scripts which yield synthetic sound output based on many of the models discussed in this book.